Discover, geometrically prove, and apply the Pythagorean Theorem.
Imagine you are helping an architect build a scale model of a sculpture. One triangular frame is a right triangle with legs 9 ft and 12 ft. How long is the third side?
In any right triangle with legs \(a\) and \(b\), and hypotenuse \(c\):
\[ a^2 + b^2 = c^2 \]
The hypotenuse is opposite the right angle, so it is always the longest side.
We usually name vertices with uppercase letters and opposite sides with matching lowercase letters.
Example. If \(a=9\) and \(b=12\), find \(c\).
\[ \begin{aligned} c &= \sqrt{9^2 + 12^2} \\ &= \sqrt{81 + 144} \\ &= \sqrt{225} \\ &= 15 \end{aligned} \]
The third side is 15 ft.
Example. If \(c=13\) and \(a=5\), find \(b\).
\[ \begin{aligned} b &= \sqrt{c^2 - a^2} \\ &= \sqrt{13^2 - 5^2} \\ &= \sqrt{169 - 25} \\ &= \sqrt{144} \\ &= 12 \end{aligned} \]
Example. If \(c=10\) and \(a=2\), find \(b\).
\[ \begin{aligned} b &= \sqrt{10^2 - 2^2} \\ &= \sqrt{100 - 4} \\ &= \sqrt{96} \\ &= \sqrt{16 \cdot 6} \\ &= 4\sqrt{6} \end{aligned} \]
When solving \(c^2=289\), the equation has two numerical roots (\(\pm 17\)), but side lengths in geometry are nonnegative. So the triangle side is \(17\), not \(-17\).
If three side lengths satisfy \(a^2+b^2=c^2\), then the triangle is right-angled.
Example: \(7,24,25\) because \(7^2+24^2=49+576=625=25^2\).
Distance between points \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
\[ AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]
This is Pythagoras applied to horizontal and vertical change.